Vertex-reinforced random walk on \(\mathbb Z\) has finite range
The stochastic process called vertex-reinforced random walk has been defined by the first named author [Ann. Probab. 16, No. 3, 1229-1241 (1988; Zbl 0648.60077)]. In the present paper this process is considered in the case where the underlying graph is \(\mathbb Z\). The authors show that the range is almost surely finite, that at least five points are visited infinitely often almost surely, and that with positive probability the range contains exactly five points. There are always points visited infinitely often but at a set of times of zero density, and the number of visits to such a point by time \(n\) may be asymptotically \(n^\alpha\) for a dense set of values \(\alpha\in(0,1)\). The power law analysis relies on analysis of a related urn model generalizing both Pólya's and Friedman's urn.
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